Finding co-ordinates and t-value for closest point between two parametric curves.

Question

Find for the two parametric equations:

$x_t1=10402 cos⁡(t/980) $

$ y_t2=11066 sin⁡(t/980)-t^2/(4.55×10^6 )$

$x_t2=11258 cos⁡(t/1120)$

$y_t2=10398 sin⁡(t/1120)$

Where t is time passed in seconds

The point where the two orbiting objects are closest is needed to be found, along with the value for time at this point.

I really don't know where to start with this problem, so any help would be greatly appreciated.

Thanks-

Answer 1

Easier if you consider slopes at first symbolically

$$ (x,y)= ( a \cos pt, b \sin pt - c t^2) $$

$$ (X,Y)= ( c \cos qt, d \sin qt),\,$$

EDIT1:

Differentiate $(X-x)^2+ (Y-y)^2 $ solve for $t$ after simplifying. Check if it gets transcendental. Then insert values for a numerical solution.